Calculus Ch5 Review – Assignment p. 401 10,16,18,39,65
The natural logarithmic function is defined by
x
1
ln x   dt
t
1
for x > 0
The domain of the natural logarithmic function is the set of all positive real numbers.
Theorem 5.1 Propoerties of the natural logarithmic function
1. The domain is (0,  ) and the range is (-  ,  )
2. The function is continuous, increasing, and one-to-one
3. The graph is concave downward
Theorem 5.2 Logarithmic Properties
If a and b are positive numbers and n is rational, then the following properties are true.
1. ln(1) = 0
2. ln(ab) = ln a + ln b
3. ln(an)=nln a
4. .ln (a/b) = ln a – ln b
The letter e denotes the positive real number such that
e
1
ln e   dt
t
1
Theorem 5.3 Derivative of the Natural Logarithmic Function
d
1
ln x   for x > 0
dx
x
Theorem 5.5 Log Rule for Integration
1
 xdx  ln x  C
Definition of Inverse Function
A function g is the inverse function of f if f(g(x) = x for each x in the domain of g and
g(f(x)) = x for each x in the domain of f. The function g is denoted by f-1
Theorem 5.6: The graph of f contains the point (a,b) if and only if the grph of f-1 contains the
point (b,a).
Theorem 5.7 The Existence of an Inverse Function
1. A function has an inverse function if and only if it is one-to-one
2. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an
inverse function.
Guidelines for Finding an Inverse Function
1. Use Theorem 5.7 to determine whether the function has an inverse function.
2. Interchange x and y and solve for y which is f-1(x).
3. Define the domain of f-1 as the range of f.
Theorem 5.8: Let f be a function whose domain is an interval. If f has an inverse function, then
the following statements are true.
1. If f is continuous on its domain, then f-1 is continuous on its domain.
2. If f is increasing on its domain, then f-1 is increasing on its domain.
3. If f is decreasing on its domain, then f-1is decreasing on its domain.
4. If f is differentiable on an interval containing c and f’(c) ≠ 0, then f-1 is differentiable at f(c).
Theorem 5.9: The derivative of an inverse function
Let f be a function that is differentiable on an interval. If f has an inverse function g, then g is
differentiable at any x for which f’(g(x)) ≠ 0. Moreover
g '( x) 
1
f '( g ( x))
Definition of the Natural Exponential Function
The inverse function of the natural logarithmic function f(x) = ln(x) is called the natural
exponential function and is denoted by f-1(x) = ex. That is, y = ex if and only if x = ln y
Properties of the Natural Exponential Function
1. The domain of f(x) = ex is (, ) and the range is (0, )
2. The function f(x) = ex is continuous, increasing, and one-to-one on its entire domain
3. The graph of f(x) = ex is concave upward on its entire domain.
4. lim e x  0 and lim e x  
x 
x
Theorem 5.11 Derivative of the natural exponential function
d x
e   e x
dx
Theorem 5.12 Integration Rules for Exponential Function
x
x
 e dx  e  C
Definition of Exponential Function to Base a
If a is a positive number (a≠1) and x is any real number, then the exponential function to the base
a is denoted by ax and is defined by
a x  e(ln a ) x
If a = 1, then y = 1x = 1 is a constant function
Definition of Logarithmic Function to Base a
If a is a positive number (a≠1) and x is any positive real number, then the logarithmic function to
the base a is denoted by
1
log a x 
ln x
ln a
Properties of Inverse Functions
1. y  a x if and only if x  log a y
2. a log a x  x for x > 0
3. log a a x  x for all x
Theorem 5.13 Derivatives for Bases Other Than e
Let a be a positive real number (a≠1) and let u be a differentiable function of x.
1.
d
 a x   (ln a )a x
dx
2.
d
du
 a u   (ln a )a u
dx
dx
3.
d
1
log a x 
dx
(ln a) x
4.
d
1 du
log a u  
dx
(ln a)u dx
Theorem 5.15
 1
 x 1 
lim 1    lim 
 e
x 
 x  x  x 
x
x
Compound Interest Formulas
Let P = amount of deposit, t = number of years, A = balance after t years, r = annual interest rate
(decimal form), and n = number of compounding per year.
 r
1. Compounded n times per year: A  P  1  
 n
2. Compounded continuously: A  Pe rt
nt